Hartry Field writes:
>The sort of theory I had in mind as a candidate for "our maximal theory" was
>ZFC with ones favorite large cardinal axioms etc. (call this ZFC+),
>supplemented with the sort of axiomatized truth theory discussed in Friedman
>and Sheard's "An axiomatic Approach to Self-Referential Truth".
So a candidate for "the most powerful recursively axiomatized first
order mathematical theory we accept", as you imagine it, is an extension
of ZFC by large cardinal axioms together with a truth theory such as
discussed in the paper by Friedman and Sheard.
Speaking only for myself, such a theory is not at all a candidate
for "the most powerful recursively axiomatized first order theory I
accept", for two reasons. First, although I do accept many large
cardinal axioms in the sense of finding them intuitively convincing
and believing their arithmetical consequences to be true, there isn't
any strongest such extension of ZFC that I accept. Saying that a
particular extension of ZFC is "a candidate" for "the most powerful
acceptable extension of ZFC by large cardinal axioms" is on the face
of it no more justified than saying that a particular natural number
is "a candidate" for "the largest natural number". Second, of the
nine maximal consistent subsets of various (jointly inconsistent)
possible axioms for a truth predicate presented by Friedman and Sheard
there is not one that captures any informal truth predicate that I
actually use in mathematical thinking or reasoning. And indeed
Friedman and Sheard themselves don't invoke any such informal understanding
of truth to justify any of these nine maximal subsets, but establish
their consistency by using ordinary mathematical models.
It may of course be that the outlook of the mathematical community
as a whole nevertheless is such that an extension of ZFC by large
cardinal axioms (it's unclear what the 'etc' refers to) together with
the sort of axiomatized truth theory discussed by Friedman and Sheard
may reasonably be called a candidate for the most powerful recursively
axiomatized first order mathematical theory that we accept, and that
provability in every such theory yields a useful notion of
"determinately true" for arithmetical statements. But a justification
for this idea is lacking. Where and how is this palpably indeterminate
notion of "determinately true" used in any mathematical thinking? If
it isn't in fact used, what is the justification for seeking the
determinacy of arithmetical truth in this conceptually and
epistemologically problematic bundle of hypothetical, indeterminate,
and sketchily invoked first order theories?
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Torkel Franzen